wavelength, A ^ , of the divergent waves also vanishes at 

 the track of the ship, as may be seen from figures 6 and 

 7. Therefore, the divergent waves can theoretically become 

 infinitely steep at the track of the ship. More precisely, 

 the steepness, s^ = a^/A^, of the divergent waves is 

 unbounded at the track of the ship if condition (36) is 

 satisfied. Both conditions (37) and (36) can be satisfied 

 simultaneously if condition (39) is satisfied, where the far- 

 field wave-amplitude function is of order l/t'' as t — °°. 

 Conditions (37) and (36), and consequently also condition 

 (39), are satisfied in the cases of the thin-ship and the 

 slender-ship approximations K,^ and Kq for the simple 

 ship form considered in the study, as may be verified 

 from figures 15 and 16 where a^(o) -► and s^((») -» °o 

 as o -* 0. 



Infinitely-steep water waves cannot exist in reality. 

 Indeed, there exists a theoretical upper bound for the 

 steepness of water waves in deep water, which is 

 approximately equal to 1/7. Condition (39) thus suggests, 

 that no divergent waves can exist within a certain region 

 in the vicinity of the track of the ship, and that the Kelvin 

 wake contains three distinct regions: (i) an inner region 

 adjacent to the track of the ship where only transverse 

 waves can exist, (ii) an outer region where both transverse 

 and divergent waves are present, and (iii) an intermediate 

 region at the boundary between the inner and outer 

 regions where steep short divergent waves, as well as 

 transverse waves, can be found. 



Surface-tension and nonlinear effects have been 

 ignored in the analysis presented in this study. This linear 

 no-surface-tension analysis predicts extremely short and 

 steep waves in the vicinity of the track of the ship. Both 

 surface-tension and nonlinear effects therefore are liable to 

 be significant, and these effects should be taken into 

 account. A linear analysis including surface-tension effects 

 should be performed first, since it is evident from the 

 results obtained in the present study and from the brief 

 description of the effects of surface tension upon the 

 Kelvin wake given in Lamb [9, pp. 468-470) and 

 Wehausen and Laitone [10, pp. 636-637] that the system 

 of divergent waves in the vicinity of the track of the ship 

 is likely to be profoundly affected by surface tension. 



It was previously found by Scragg [5] that, for a 

 ship bow form with a large flare angle, the zeroth-order 

 slender-ship approximation KQ(t) predicts a sharp peak in 

 the value of the amplitude of the divergent waves at a 



value of cr equal to approximately half the entrance angle 

 p. This finding of Scragg has been verified in this study, 

 as may be seen from figures 15, 16 and 18. Furthermore, 

 the magnitude of the steepness of the divergent waves has 

 been found to increase very rapidly as the Froude number 

 decreases below a certain threshold value, as is shewn in 

 figure 19. 



The line along which the steepness s ^ (o) of the 

 divergent waves has a peak and the lines along which 

 s^(o) takes the large values 1/7, 1/15 and 1/20 have been 

 determined, for a simple ship form, on the basis of both 

 the zeroth-order slender-ship approximation KQ(t) and the 

 Michell thin-ship approximation K^(t). Figure 20, 

 corresponding to the slender-ship approximation KQ(t), 

 shows that these lines are well inside the Kelvin angle, and 

 that the large-steepness lines are much closer to the track 

 of the ship than the line corresponding to the f)eak in the 

 steepness of the divergent waves. 



The lines, depicted in figures 20 and 21, along 

 which the steepness of the divergent waves takes large 

 constant values are independent of the value of the 

 Froude number, but they strongly depend on the hull 

 shape, as may be seen from figure 22 where "constant- 

 steepness lines" corresponding to several values of the 

 entrance angle p and of the flare angle y are depicted. 

 This figure shows that the short divergent waves in the 

 Kelvin wake become steeper as the entrance angle 

 increases and/or as the flare angle decreases. 



It was found that the lines, along which the 

 steepness of the divergent waves takes large constant 

 values, predicted by the slender-ship approximation KfjlX) 

 and the thin-ship approximation K^(t) are quite different 

 from one another, as may be seen from figure 21. This 

 figure therefore indicates the need for performing 

 additional calculations based on a more realistic 

 mathematical model than the simple thin-ship and slender- 

 ship approximations used in this study. These two 

 approximations correspond to simple special cases of the 

 Neumann-Kelvin theory, which should then be used. In 

 particular, it would be useful to determine whether this 

 more realistic theory predicts that the steepness of the 

 divergent waves in the Kelvin wake exhibits a peak (or 

 several peaks), as was found by using the slender-ship 

 approximation KQ(t) for a ship bow form with large flare 

 angle. Figure 21 specifically demonstrates the importance 

 of obtaining accurate predictions of the far-field wave- 



18 



